D. Zack Garza

- You run into a “space” in the wild. Which one is it?
- How many possible spaces
*could*it be? - How much information is needed to specify our space uniquely?

- Possible to fit data to a high-dimensional manifold
- Makes clustering/grouping easier
*(Here, slice with a hyperplane)*

- Extract info about an entire family of objects and how they vary.
- Also useful for outlier detection!

What structure does your space have?

What can you “measure” locally?

How might it vary in ways you

*can’t*measure?Important question before attempting to classify:

**What does “space” mean**?

- Need to pick a category to work in.

**What does “which” mean**?

- How to distinguish? Need an equivalence relation!

- Plan: compare classification theorem in topology and algebraic geometry
- Hopefully see some fun spaces along the way!
- But first: address classification and the notion of “sameness”

Early classification efforts:

**conic sections**.- Apollonius, 190 BC, Ancient Greeks

Key idea: realize as intersection loci in bigger space

- (Projectivize \(\mathbb{R}^2\).)

- A conic is specified by 6 parameters:

\[\begin{align*} A x^{2}+B x y+C y^{2}+D x+E y+F=0 \\ \\ Q :=\left(\begin{array}{ccc} A & B / 2 & D / 2 \\ B / 2 & C & E / 2 \\ D / 2 & E / 2 & F \end{array}\right), \quad \mathbf{x} = [x, y, 1] \\ \implies \mathbf{x}^t Q \mathbf{x} = 0 \end{align*}\]

\(\det(Q)\) | Conic |
---|---|

\(<0\) | Hyperbola |

\(=0\) | Parabola |

\(>0\) | Ellipse |

Each conic is a variety

- Can obtain
*every*conic by "modulating* 6 parameters.

- Can obtain
\(\mathbb{R}^6\): too much information: scaling by a nonzero \(\lambda \in \mathbb{R}\) yields the same conic, so reduce the space \[ [A, B, C, D, E, F]\in \mathbb{R}^6 \mapsto [A: B: C: D: E: F] \in \mathbb{RP}^5 .\]

Important point: \(\mathbb{RP}^5\) is a projective variety and a smooth manifold! Tools available:

- Dimension (what does a generic point look like?)
- Tangent and cotangent spaces, differential forms
- Measures, metrics, volumes, integrals
- Intersection theory (Bezout’s Theorem!), subvarieties, curves
- Linear algebra and Combinatorics (enumerative questions)

- We can imagine a
*moduli space of conics*that parameterizes these:

Some Calc III review:

(discriminants), the equation becomes \(\mathbf x^t E \mathbf x = 0\) and we have a classification:

What is the moduli space? It sits inside \(\mathbb{R}^{10}\), possibly \(\mathbb{RP}^{9}\) but not in the literature!

- Problem: infinitely many points in these moduli spaces correspond to the same “class” of conic

How to address: Klein’s Erlangen program, understand the geometry of a space by understanding its structure-preserving automorphisms.

For topological spaces: a Lie group acting on the space.

Can then “mod out” by the appropriate morphisms to (hopefully) get finitely many equivalence classes

**Algebraic Variety**: Irreducible ,zero locus of some family \(f\in \mathbb{k}[x_1, \cdots, x_n]\) in \(\mathbb{A}^n/\mathbb{k}\).- Equivalently, a locally ringed space \((X, \mathcal{O}_X)\) where \(\mathcal{O}_X\) is a sheaf of finite rational maps to \(\mathbb{k}\).

**Projective Variety**: Irreducible zero locus of some family \(f_n \subset \mathbb{k}[x_0, \cdots, x_n]\) in \(\mathbb{P}^n/\mathbb{k}\)- Admits an embedding into \(\mathbb{P}^\infty/\mathbb{k}\) as a closed subvariety.

**Dimension**of a variety: the \(n\) appearing above.

**Topological Manifold**: Hausdorff, 2nd Countable, topological space, locally homeomorphic to \(\mathbb{R}^n\).- Equivalently, a locally ringed space where \(\mathcal{O}_X\) is a sheaf of continuous maps to \(\mathbb{R}^n\).

**Smooth Manifold**: Topological manifold with a smooth structure (maximal smooth atlas) with \(C^\infty\) transition functions.- Equivalently, a locally ringed space where \(\mathcal{O}_X\) is a sheaf of smooth maps to \(\mathbb{R}^n\).

**Algebraic Manifold**: A manifold that is also a variety, i.e. cut out by polynomial equations. Example: \(S^n\).

- Manifolds will be compact and without boundary, varieties are (probably) smooth, separated, of finite type.

- Pick a category, understand
*all*of the objects (identifying a moduli “space”) and*all*of the maps.- Understand all topological spaces up to ???
- Homeomorphism?
- Diffeomorphism?
- Homotopy-Equivalence?
- Cobordism?

- Understand all algebraic and/or projective varieties up to
- Biregular maps?
- Birational maps?
- Locally ringed morphisms?

- Understand all topological spaces up to ???

Classifying manifolds up to homeomorphism: stratify “moduli space” of topological manifolds by dimension.

Dimensions 0,1,2,3:

- Smooth = Top. See smooth classification.

- Dimension 4:
*Topologically*classified by surgery, but barely, and not smoothly.

- Dimension \(n\geq 5\):
- Uniformly “classified” by surgery, s-cobordism, with a caveat:
- \(\pi_1\) can be any finitely presented group – word problem
- Instead, breaks homotopy type of a fixed manifold up into homeomorphism classes

Generally expect things to split into more classes.

- Dimension 0: The point (terminal object)
- Dimension 1: \(\mathbb{S}^1, \mathbb{R}^1\)

- Dimension 2: \(\left\langle\mathbb{S}^2, \mathbb{T}^2, \mathbb{RP}^2 \mid \mathbb{S}^2 = 0,\,\,3\mathbb{RP}^2 = \mathbb{RP}^2 + \mathbb{T}^2 \right\rangle\).
- Classified by \(\pi_1\) (orientability and “genus”). Riemann, Poincaré, Klein.

- Dimension 2: closed + orientable \(\implies\) complex
**Uniformization**: Holomorphically equivalent to a quotient of one of three spaces/geometries:- \(\mathbb{CP}^1\), positive curvature (spherical)

- \(\mathbb{C}\), zero curvature (flat, Euclidean)
- \(\mathbb{H}\) (equiv. \(\mathbb{D}^\circ\)), negative curvature (hyperbolic)

- \(\mathbb{CP}^1\), positive curvature (spherical)
- Stratified by genus
- Genus 0: Only \(\mathbb{CP}^1\)
- Genus 1: All of the form \(\mathbb{C}/\Lambda\), with a distinguished point \([0]\), i.e. an elliptic curve.
- Has a topological group structure!

- Genus \(\geq 2\): Complicated?

Doesn’t capture holomorphy type completely.

- Geometric structure: a diffeo \(M\cong \tilde M/\Gamma\) where \(\Gamma\) is a discrete Lie group acting freely/transitively on \(X\) (as in Erlangen program)
- Decompose into pieces with one of 8 geometries:
- Spherical \(\sim S^3\)
- Euclidean \(\sim \mathbb{R}^3\)
- Hyperbolic \(\sim \mathbb{H}^3\)
- \(S^2\times \mathbb{R}\)
- \(\mathbb{H}^2\times \mathbb{R}\)
- \(\widetilde{\mathrm{SL}(2, \mathbb{R})}\)
- “Nil”
- “Sol”

- Proved by Perelman 2003, Ricci flow with surgery.

- 4-manifolds: classified in the topological category by surgery, but not in the smooth category
- Hard! Will examine special cases of Calabi-Yau
- Open part of Poincaré Conjecture.

- Dimension \(\geq 5\): surgery theory, strong relation between diffeomorphic and s-cobordant.

Every smooth manifold admits a Riemannian metric, so consider Riemannian manifolds

Here \(H\leq \mathrm{SO}(n)\) is the

*holonomy*group:

- Berger’s classification for smooth Riemannian manifolds, one of 7 possibilities. \[ \tiny \begin{array}{|c|c|c|c|c|} \hline n=\operatorname{dim} M & H & \text { Parallel tensors } & \text { Name } & \text { Curvature } \\ \hline n & \mathrm{SO}(n) & g, \mu & \text {orientable} & \\ \hline 2 m(m \geq 2) & \mathrm{U}(m) & g, \omega & \textbf{Kähler} & \\ \hline 2 m(m \geq 2) & \mathrm{SU}(m) & g, \omega, \Omega & \textbf{Calabi-Yau} & \text {Ricci-flat} \\ \hline 4 m(m \geq 2) & \mathrm{Sp}(m) & g, \omega_{1}, \omega_{2}, \omega_{3}, J_{1}, J_{2}, J_{3} & \textbf{hyper-Kähler} & \text {Ricci-flat} \\ \hline 4 m(m \geq 2) & (\mathrm{Sp}(m) \times \mathrm{Sp}(1)) / \mathbb{Z}_{2} & g, \Upsilon & \text {quaternionic-Kähler} & \text {Einstein} \\ \hline 7 & \mathrm{G}_{2} & g, \varphi, \psi & \mathrm{G}_{2} & \text {Ricci-flat} \\ \hline 8 & \operatorname{Spin}(7) & g, \Phi & \operatorname{Spin}(7) & \text {Ricci-flat} \\ \hline \end{array} \]

Types in bold: amenable to Algebraic Geometry. \(G2\) shows up in Physics!

- Ricci-flat, i.e. Ricci curvature tensor vanishes
*(Measures deviation of volumes of “geodesic balls” from Euclidean balls of the same radius)*

Work over \(\mathbb{C}\) for simplicity, take all dimensions over \(\mathbb{C}\).

Minimal model program: classifying complex projective varieties.

Stratify the “moduli space” of varieties by \(\mathbb{k}-\)dimension.

Dimension 1:

- Smooth Algebraic curves = compact Riemann surfaces, classified by genus
- Roughly known by Riemann: moduli space of smooth projective curves \(\mathcal{M}_g\) is a connected open subset of a projective variety of dimension \(3g-3\).

We’ll come back to these!

- Dimension 2:
Smooth Algebraic Surfaces: Hard. See Enriques classification.

Setting of classical theorem: always 27 lines on a cubic surface!

Example Clebsch surface, satisfies the system \[ \begin{array}{l} x_{0}+x_{1}+x_{2}+x_{3}+x_{4}=0 \\ \\ x_{0}^{3}+x_{1}^{3}+x_{2}^{3}+x_{3}^{3}+x_{4}^{3}=0 \end{array} \]

- Equivalently, Riemann surfaces with one marked point.
- Equivalently, \(\mathbb{C}/\Lambda\) a lattice, where homothetic lattices (multiplication by \(\lambda \in \mathbb{C}- \{0\}\)) are equivalent.
- Generalize to \(\mathbb{C}^n/\Lambda\) to obtain
*abelian varieties*.

- \(\mathcal{M}_g\): the moduli space of compact Riemann surfaces (curves) of genus \(g\), i.e. elliptic curves.

Definition:

**Kodaira Dimension**- \(X\) has some canonical sheaf \(\omega_X\), you can take some sheaf cohomology and get a sequence of integers (
*plurigenera*)

- \(X\) has some canonical sheaf \(\omega_X\), you can take some sheaf cohomology and get a sequence of integers (

\[\begin{align*} P_{\mathbf{n}} (X) &:= h^0(X, \omega_X^{\otimes \mathbf{n}}) \quad n\in \mathbb{Z}^{\geq 0} \\ \\ \implies \kappa(X) &:= \limsup_{\mathbf n \to \infty} {\log P_{\mathbf n}(X) \over \log(\mathbf{n}) } \end{align*}\]

Every such surface has a minimal model of one of 10 types:

\(\kappa = -\infty\) (2 main types)

- Rational: \(\cong \mathbb{CP}^2\)

- Ruled: \(\cong X\) for \(\mathbb{CP}^1 \to X \to C\) a bundle over a curve.

- Called “ruled” because every point is on some \(\mathbb{CP}^1\).

- Type VII

\(\kappa = 0\) (Elliptic-ish, 4 types)

- Enriques (all (quasi)-elliptic fibrations)

- Hyperelliptic

- Taking Albanese embedding (generalizes Jacobian for curves) yields an elliptic fibration
*(i.e. a surface bundle, potentially with singular fibers)*

- \(K3\) (Kummer-Kahler-Kodaira) surfaces

- Toric and Abelian Surfaces:

- 2 dimensional abelian varieties (projective algebraic variety + algebraic group structure).
- Compare to 1 dimensional case: all 1d complex torii are algebraic varieties,
- Riemann discovered that most 2d torii are
*not*.

- Riemann discovered that most 2d torii are

- Kodaira Surfaces

\(\kappa = 1\): Other elliptic surfaces

- Properly quasi-elliptic.

Elliptic fibration, but almost all fibers have a node.

\(\kappa = 2\) (Max possible, “everything else”)

- General type

- Definitions:
Define a

*complex torus*as \(\mathbb{T} = (\mathbb{C}^{\times})^n \subseteq \mathbb{C}^n\)Can be written as the zero set of some \(f\in \mathbb{C}[x_0, \cdots, x_n]\) in \(\mathbb{C}^{n+1}\).

Generalizes to algebraic groups over a field: \((\mathbb{G}_m)^n\) (analogy: maximal torus/Cartan subalgebra in Lie theory)

**Toric variety**: \(X\) contains a dense Zariski-open torus \(\mathbb{T}\), where the action of \(\mathbb{T}\) on itself as a group extends to \(X\).

- Flavor: spaces modeled on convex polyhedra
- Examples: bundles over \(\mathbb{CP}^n\).
- Why study:
- Model spaces by rigid geometry, generalize things like Bezier curves
- Some are determined by rigid combinatorial data (“fan”, or polytopes)
- Combinatorial data for constructions in mirror symmetry, e.g. Calabi-Yaus (1/2 of one billion threefolds!)

- Model spaces by rigid geometry, generalize things like Bezier curves

- As complex manifolds:
- A symplectic manifold \((X, \omega)\) with an integrable almost-complex structure \(J\) compatible with \(\omega\).
- Yields an inner product on tangent vectors: \(g(u, v) := \omega(u, Jv)\) (i.e. a metric)

- Examples:
*all*smooth complex projective varieties- But not all complex manifolds (exception: Stein manifolds)

- Specialize to Calabi-Yaus: compact, Ricci-flat, trivial canonical.
- Calabi’s Conjecture and Yau’s field medal: existence of Ricci-flat Kahlers (Calabi-Yaus)

Trivial canonical \(\implies\) exists a nowhere vanishing top form = top wedge of \(T^* X\) is the trivial line bundle

- Another from Berger’s classification, special case of Kahler

- Applications: Physicists want to study \(G_2\) manifolds (an exceptional Lie group, automorphisms of octonions)
- Part of \(M\)-theory uniting several superstring theories, but no smooth or complex structures.
- Indirect approach: compactify an 11-dimension space, one small \(S^1\) dimension \(\to\) 10 dimensions
- 4 spacetime and 6 “small” Calabi-Yau

- Superstring theory: a bundle over spacetime with fibers equal to Calabi-Yaus.

Roughly, genera of fibers will correspond to families of observed particles.

- As manifolds:
Ricci-flat: vacuum solutions to (analogs of) Einstein’s equations with zero cosmological constant

Setting for mirror symmetry: the symplectic geometry of a Calabi-Yau is “the same” as the complex geometry of its mirror.

Yau, Fields Medal 1982: There are Ricci flat but non-flat (nontrivial holonomy) projective complex manifolds of dimensions \(\geq 2\).

As varieties: the canonical bundle \(\Lambda^n T^* V\) is trivial

Compact classification for \(\mathbb{C}-\)dimension:

- Dimension 1: 1 type, all elliptic curves (up to homeomorphism)
- Dimension 2: 1 type, \(K3\) surfaces

- Dimension 3: (threefolds) conjectured to be a bounded number, but unknown.
- At least 473,800,776!